Covariances of zero crossings in gaussian processes

For a zero-mean Gaussian process, the covariances of zero crossings can be expressed as the sum of quadrivariate normal orthant probabilities. In this paper, we demonstrate the evaluation of zero crossing covariances using one-dimensional integrals. Furthermore, we provide asymptotics of zero crossing covariances for large time lags and derive bounds and approximations. Based on these results, we analyze the variance of the empirical zero crossing rate. We illustrate the applications of our results by autoregressive (AR) fractional Gaussian noise and fractionally integrated autoregressive moving average (FARIMA) processes.